Estimating the distribution of marks of a homogeneous marked Poisson process

In this paper we propose an estimator of the distribution of events of different kinds in a homogeneous Poisson process. We give an explicit solution for the maximum likelihood estimator of the distribution and derive its strong consistency and asymptotic normality. We also provide an order restricted estimator of the distribution and derive its consistency and asymptotic distribution. The inference problem gives rise to a Sylvester-Ramanujan system of equations. We discuss application of the estimator to the detection of neutrons in a novel detector developed at the European Spallation Source in Lund, Sweden.